PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, SERIES B Volume 7, Pages 183–201 (November 25, 2020) https://doi.org/10.1090/bproc/53
GRASSMANN SEMIALGEBRAS AND THE CAYLEY-HAMILTON THEOREM
LETTERIO GATTO AND LOUIS ROWEN
(Communicated by Jerzy Weyman)
Abstract. We develop a theory of Grassmann semialgebra triples using Hasse- Schmidt derivations, which formally generalizes results such as the Cayley- Hamilton theorem in linear algebra, thereby providing a unified approach to classical linear algebra and tropical algebra.
1. Introduction The main goal of this paper is to define and explore the semiring version of the theory [4–6] of the first author concerning the Grassmann exterior algebra, viewed more generally in terms of negation maps and systems, continuing the approach of [21]. This provides a robust structure which unifies and generalizes four major versions of Grassmann structures, cf. Note 1.1, and provides a generalization of the Cayley-Hamilton theorem in Theorem 3.15. In the process we investigate Hasse- Schmidt derivations on Grassmann systems. The version given in Theorem 2.4 (over a free module V over an arbitrary semifield) is the construction which seems to “work”. Generalizing negation in Definition 1.3 to the notion of a “negation map” (−) satisfying all the usual properties of negation except a(−)a = 0 (studied in [21]), we find that the Grassmann semialgebra of a free module V , described in Theorem 2.4, has a natural negation map on all homogeneous vectors, with the ironic exception of V itself, obtained by switching two tensor components. ⊗(n) − Theorem A (Theorem 2.4). n≥2 V has a negation map ( ) satisfying ···¯ − π ···¯ bπ(i1) bπ(it) =( ) bi1 bit , ∈ for bij V.
Received by the editors May 14, 2018, and, in revised form, May 13, 2020. 2020 Mathematics Subject Classification. Primary 15A75, 16Y60, 15A18; Secondary 12K10, 14T10. Key words and phrases. Cayley-Hamilton theorem, exterior semialgebras, Grassmann semial- gebras, Hasse-Schmidt derivations, differentials, eigenvalues, eigenvectors Laurent series, Newton’s formulas, power series, semifields, systems, semialgebras, tropical algebra, triples. The first author was partially supported by INDAM-GNSAGA and by PRIN ”Geometria sulle variet`a algebriche” Progetto di Eccellenza Dipartimento di Scienze Matematiche, 2018–2022 no. E11G18000350001. The second author was supported in part by the Israel Science Foundation, grant No. 1207/12 and his visit to Torino was supported by the “Finanziamento Diffuso della Ricerca”, grant no. 53 RBA17GATLET, of Politecnico di Torino.
c 2020 by the authors under Creative Commons Attribution-Noncommercial 3.0 License (CC BY NC 3.0) 183 184 LETTERIO GATTO AND LOUIS ROWEN
This provides us “enough” negation, coupled with the relation ◦ of Defini- tion 1.6 (designed to replace equality), to adapt the methods of [6] to negation maps and systems of [21] to study T (V ), with T the nonzero simple tensors. One obtains a Grassmann algebra by modding out all elements of the form v ⊗v, v ∈ V . A weaker version is obtained when we take a given base {b0,b1,...,bn−1} of V and mod only the bi ⊗ bi, 0 ≤ i
Theorem C (Theorem 3.12). The polynomial D{z} is a quasi-inverse of D{z}, in the sense that D{z}D{z}u u for all u ∈ Vn. Our main result, Theorem 3.14, describes the relation between a Hasse-Schmidt derivation D{z} and its “quasi-inverse” D{z} defined in such a way to yield: Theorem D (Theorem 3.14). D{z}(D{z}u ∧ v) u ∧ D{z}v. In the classical case, one gets equality as shown in Remark 3.16, so we recover [6]. Our main application in this paper is a generalization of the Cayley-Hamilton theorem to semi-algebras. Theorem E (Theorem 3.15). (1.1) ··· ∧ − ··· ∧ ((Dnu + e1Dn−1u + + enu) v)( )((Dnu + e1Dn−1u + + enu) v) 0 >0 for all u ∈ Vn. We obtain true equality in these theorems by modding out all elements of the form v ⊗ v. Again we recover [6] in the classical case. GRASSMANN SEMIALGEBRAS AND THE CAYLEY-HAMILTON THEOREM 185
1.1. Basic notions. Much of this section is a review of [21], as summarized in [22], and also as in [15]. As customary, N denotes the natural numbers including 0, N+ denotes N \{0}, Q denotes the rational numbers, and R denotes the real numbers, all ordered monoids under addition. A semiring†† (A, +, ·) is an additive abelian semigroup (A, +) and multiplicative semigroup (A, ·) satisfying the usual distributive laws. A semiring† (A, +, ·, 1)is a semiring†† with a multiplicative unit 1. (Thus, an ideal of a semiring† is a semiring††.) A semifield† is a semiring† whose multiplicative monoid is a group. A semiring (resp. semifield) [9] is a semiring† (resp. semifield†) with an absorbing element 0 formally adjoined. Definition 1.2. A T -module over a set T is an additive monoid (A, +, 0)witha scalar multiplication T×A→Asatisfying the following axioms, ∀k ∈ N,a∈T, b, b ∈A: j T k k (i) (Distributivity over ): a( j=1 bj )= j=1(abj ). (ii) a0A = 0A. A T -monoid module over a multiplicative monoid T is a T -module satisfying the extra conditions
1T b = b, (a1a2)b = a1(a2b), ∀ai ∈T,b∈A. A T -semiring is a semiring that is also a T -monoid module over a given mul- tiplicative submonoid T . This paper only concerns T -semirings, which are closely related to blueprints in [19]. We put T0 = T∪{0}. Tensor products over semirings [17,18,23] are analogous to tensor products over rings. The tensor product M1 ⊗A M2 of right and left A-modules M1 and M2 is (F1 ⊕F2)/Φ, where Fi is the free module (respectively right or left) with base Mi, and Φ is the congruence generated by all (1.2) x1,j , x2,k , x1,j ,x2,k , j k j,k
(x1a, x2), (x1,ax2) ∀xi,xi,j ,xi,k ∈Mi,a∈A.
1.2. Negation maps, triples, and systems. Definition 1.3. A negation map on a T -module M over a given set T is a semigroup isomorphism (−):M → M of order ≤ 2, written b → (−)b,whichalso respects the T -action in the sense that (−)(ab)=a((−)b) for a ∈T,b∈ M. A semiring†† negation map on a semiring†† A is a negation map which satisfies (−)(ab)=a((−)b)=((−)a)b for all a, b ∈A. In the classical case the negative is a negation map. For tropical algebra, one could just take (−) to be the identity map, but we want a less trivial example. We write a(−)b for a +((−)b), (±)a for {a, (−)a}, and a◦ for a(−)a, called a quasi-zero.ThesetM ◦ of quasi-zeroes is an important T -submodule of M.When A is a semiring, A◦ is an ideal. We define (−)0a to be a and, for k ∈ N, we inductively define (−)ka to be (−)((−)k−1a). 186 LETTERIO GATTO AND LOUIS ROWEN
Lemma 1.4. ((−)ka)((−)k a)=(−)k+k (aa) for a, a ∈A.
Proof. ((−)ka)((−)k a)=(−)((−)k−1a)((−)k a)=(−)((−)k+k −1(aa)) =(−)k+k (aa), by induction on k. Definition 1.5. A pseudo-triple is a collection (A, T , (−)), where (−) is a nega- tion map on both T and A,andA is a (T , (−))-module. A TA-pseudo-triple (A, TA, (−)) is a T -module A, with TA designated as a distinguished subset, and a negation map (−) satisfying (−)TA = TA. A TA-triple, ◦ called a triple when T is understood, is a TA-pseudo-triple, in which TA ∩A = ∅ and TA generates (A\{0}, +). Since TA is usually clear from the context, we abuse notation and write T for ◦ TA ⊆A. A triple is uniquely negated when for any a ∈T, a + b ∈A implies b =(−)a. The structure is rounded out with the following relation.
Definition 1.6. Define the ◦-surpassing relation ◦ on a module M with nega- ◦ tion map by a0 ◦ a1 if a1 = a0 + d for some d ∈ M . A uniquely negated triple (A, T , (−)) together with a surpassing relation is called a system.1
Remark 1.7. The relation ◦ on a system restricts to equality on T ,by[21,Propo- sition 4.4]. In fact ◦ is used to replace equality when we work with triples, and identities in classical algebra can often be replaced by relations expressed in terms of ◦, by means of the transfer principle of [1], formulated for systems in [21, The- orem 6.17]. 1.3. Functions to A. The next construction, discussed in [16, §4.2], enables us to describe power series in a structural context. From now on, we suppose (S, +) is a semigroup, often (N, +, 0). Given a triple (A, T , (−)), AS denotes the maps from S to A,andT S denotes the nonzero maps of AS sending S to T . For example, for c ∈A, the constant function c˜ is given byc ˜(s)=c for all s ∈ S. We modify the definition of support from [16, Definition 4.2]. Definition 1.8. Given f ∈AS we define its support supp(f) := {s ∈ S : f(s) = 0}, and supp(AS)for∪{supp(f):f ∈AS}. Definition 1.9. AsetA of maps f : S →Ais convolution admissible if for each f,g ∈A and s ∈ S there are only finitely many s ∈ supp(f),s ∈ supp(g), with s + s = s. Example 1.10. A is convolution admissible whenever S = N(I) (the direct sum) for some index set I, since the condition of Definition 1.9 already holds in S. Definition 1.11. Suppose A is a convolution admissible set. The convolution product A ×A →A is given by defining fg to be the function satisfying fg(s)= f(s)g(s). s+s=s
1 In [16], in the systemic setting, a more general notion of surpassing map is used, and ANull ◦ ◦ is introduced which equals A when = ◦; here, using Lemma 2.18 as justification, we use A and = ◦ to avoid complications. GRASSMANN SEMIALGEBRAS AND THE CAYLEY-HAMILTON THEOREM 187
The intuitive way to receive a negation map on A from a negation map on A would be to define ((−)f)(s)=(−)(f(s)); these maps also are convolution admis- sible, so one would expand A to include them.
1.4. Graded semirings and modules. We want to grade semirings and their modules. We define direct sums in the usual way. Definition 1.12. An L-graded T -semiring† is a T -semiring† R which also is an L-graded T -module R := ⊕∈LR for semigroups (R, +) satisfying the following conditions, where T = T∩R :
(i) T = ∪∈LT; (ii) RR ⊆R+ , ∀ , ∈ L. † Note that R0 is a T0-module, and also a semiring , over which each R ∪{0} is a module. When we turn to Grassmann semialgebras, L will be ordered with a minimal >0 element 0; one could take L = N, for example. We write M := ⊕>0M,a submodule of M lacking the constant component. Then R>0 is a sub-semiring†† of R.
1.4.1. Super-semialgebras. Here is an interesting special case.
Definition 1.13. A super-semialgebra is a Z2-graded semialgebra A := A0⊕A1, i.e., satisfying twist multiplication: (1.3) (a0,a1)(a0,a1)=(a0a0 + a1a1,a0a1 + a1a0).
A natural way of getting a Z2-grade from an N-graded semialgebra is to take the 0-grade to be the set of even indices and the 1-grade to be the set of odd indices.
1.4.2. The power series semiring of a graded semiring†. From now on, we take S = N as in Example 1.10, which is equivalent to the following. Definition 1.14. Given an N-graded T -semiring R with respect to the semigroup (N, +), we define the power series semiring R[[z]] over a central indeterminate z,in the usual way as possibly infinite formal sums (with convolution product), and its R R j sub-semiring [z]= j j z . Lemma 1.15. R[[z]] and R[z] are indeed semirings. Both R[[z]] and R[z] are graded by the powers of z. Proof. R[[z]] satisfies the axioms of a semiring, by the customary verification, and its subset R[z] is closed under addition and multiplication, so both are semirings. j k j+k The last assertion follows from the fact that Rj z Rkz ⊆Rj+kz .
Definition 1.16. Suppose A is a semialgebra over a commutative base semiring†. We write End(A) for the set of module maps A→A. Given D ∈ End(A)S, we write Ds for the map given by s → D(s)ands → 0, ∀s = s. In the other direction, given fs :∈ End(A), s ∈ S, each of singleton support {s} with the {s} distinct, f S f define D ∈ End(A) via D (s)=fs. Remark 1.17. Df (s)(ab)=Df (s)(a)Df (s)(b), ∀a, b ∈A, under the usual product, seen by matching terms in the left side and the right side. 188 LETTERIO GATTO AND LOUIS ROWEN
1.5. Higher derivations. This discussion applies to any semialgebra A, not nec- essarily associative and not necessarily a triple. A derivation δ : A→Ais a map in End(A) satisfying δ(ab)=aδ(b)+δ(a)b. The following concepts were introduced by Hasse and Schmidt [10] and studied further by Heerema [11–13]. For S convolution admissible (not necessarily associative), a homogeneous map A S D in (End ) is called a higher derivation of A if it satisfies the conditions: ∀ ∈ ∀ ∈A (i) Ds(ab)= s+s=s Ds (a)Ds (b), s S, a, b . (ii) D0 = 1 (the identity map on A). Property (i) is called the Leibniz rule, obtained from the more familiar Leibniz rule for derivations for S = N by dividing by k!. [10, pp. 190-191] indicates how to define a higher derivation D. Wehavea somewhat different take, along classical lines. We consider semialgebras over semi- fields containing Q>0, for the following definition to make sense. Given a map A→A f j A→A f : [[z]] we define its exponential exp(f)= j≥1 j! : [[z]]. It is well-known that the exponential of a derivation is a homomorphism. In fact we have: k Lemma 1.18. If d1,d2,... is a sequence of derivations, then dkz : A→A[[z]] satisfies Leibniz’ rule. Its exponential is a semialgebra homomorphism: D := r k A→A Drz :=exp( k≥1 dkz ): [[z]]. Proof. Given in [24] and [4, Propositions 3.4.2 and 3.4.3]. The proof evidently extends to semialgebras over semifields containing Q>0. Matching coefficients in Lemma 1.18, one gets precisely the Schur polynomials associated to the sequence d1,d2,....Inparticular: d2 d3 d4 1 1 D = d ,D= 1 +d ,D= 1 +d d +d ,D= 1 + d2d + d2+d d +d , 1 1 2 2 2 3 3! 1 2 3 4 4! 2 1 2 2 2 1 3 4 defines a higher derivation D. When each ds = δ for a given derivation δ,wecallD the higher derivation of δ.
2. Grassmann semialgebras Suppose A is a commutative semiring and V is an A-module. H will denote an A-semialgebra generated by V (Often H will be the tensor algebra T (V ) defined T below). We write Hk for the products of length k of elements of V ,andH≥k for T A the ideal j≥k Hj (with repetitions). Thus H = + V + H≥2. The elements of T T − kl Hk and Hl will satisfy wkwl =( 1) wlwk, leading to the subject of our study. Definition 2.1. A Grassmann,orexterior, semialgebra, over a semiring† A and an A-module V , is a semialgebra H generated by A and V ,asabove,together with a negation map on H≥2 andanassociativewedge product ∧ : H × H → H satisfying
(2.1) v1 ∧ v2 =(−)v2 ∧ v1 for vi ∈ V. π π Thus vπ(1) ∧···∧vπ(t) =(−) v1 ∧···∧vt for t ≥ 2, where (−) denotes the sign of the permutation. This ties in with the theory of triples since, taking T (V )tobe the nonzero products of elements of V ,thenfork ≥ 2, (H≥2, T (V )≥2, (−)) often is atriple,for(−) suitably defined (as in Theorem 2.4). Since T (V )≥2 ⊂ H≥2,itcan GRASSMANN SEMIALGEBRAS AND THE CAYLEY-HAMILTON THEOREM 189 be bypassed by restricting functions, but the negation map (−) will play a crucial role.
Lemma 2.2. If V is spanned by {bi : i ∈ I}, then to verify the Grassmann relation (2.1) it is enough to check that
bi ∧ bj =(−)bj ∧ bi, ∀i, j ∈ I. Proof. Distributivity yields αibi ∧ βj bj = αiβj bi ∧ bj =(−) αiβj bj ∧ bi =(−) βjbj ∧ αibi , yielding the assertion. We write vk for v ∧···∧v taken k times. 2 2 2 Lemma 2.3. ( α a ) ◦ α a for any Grassmann semialgebra. ii i i 2 2 2 ∧ ∧ Proof. ( αiai) = αi ai + i Proof. By Lemma 2.2, we may take a generating set {bi : i ∈ I} of V ,whereI is an ordered index set. We define a negation on V ⊗ V by (−)bi ⊗ bj = bj ⊗ bi. (This is possible since it preserves the bilinear relations defining the tensor product.) Since this is homogeneous of degree 2, it defines a negation on G(V )2 given by (−)bi⊗bj = bj ⊗bi. When i Lemma 2.6. If v ∈ Gi and v ∈ Gj for i, j ≥ 1 then (2.2) v ∧ v =(−)i+j v ∧ v, where (−) is given as in Theorem 2.4. Proof. Easy induction on i and j. Definition 2.7. T (V )◦ is the ideal of T (V ) generated by T (V )◦ and all elements v ⊗ v, v ∈ V. ◦ 1 ∈A ⊗ 1 ⊗ ◦ (ThisisjustT (V ) when 2 since then v v =(2 v v) ). We now weaken ◦. Definition 2.8. (Supplanting Definition 1.6) a0 a1 in T (V )ifa1 = a0 + d for some d ∈ T (V )◦. 2.0.1. The standard Grassmann semialgebra. Recall that the way to define fac- tor structures in universal algebra (in particular, for semirings† or modules over semirings†) is to mod out by a congruence. 2 Theorem 2.9. If v = 0 for all v in V , then for any permutation π and all vi ∈ V, (i) v ∧ v1 ∧···∧vn ∧ v = 0. (ii) vπ(1) ∧···∧vπ(n) = v1 ∧···∧vn if π is even; (iii) v1 ∧···∧vn + vπ(1) ∧···∧vπ(n) = 0 if π is odd. Thus the only quasi-zeros are 0. Proof. Linearizing yields 2 2 2 ∧ ∧ ∧ ∧ 0 =(v1 + v2) = v1 + v2 + v1 v2 + v2 v1 = 0 + 0 + v1 v2 + v2 v1, so v1 ∧ v2 + v2 ∧ v1 = 0. Now (i) is by induction on n,sincev ∧ v1 ∧···∧vn ∧ v = v ∧ v1 ∧···∧vn ∧ +0 =(v ∧ v1 + v1 ∧ v) ∧···∧vn ∧ v = 0. To get (ii) and (iii) we write π as a product of transpositions π1 ···πk of the form (i, i +1). If k =2thenv1 ∧ v2 ∧ v3 = v1 ∧ v2 ∧ v3 +(v2 ∧ (v1 ∧ v3 + v3 ∧ v1)) = (v1 ∧ v2 + v2 ∧ v1) ∧ v3 + v2 ∧ v3 ∧ v1 = v2 ∧ v3 ∧ v1, and then we have (ii) for all even k. For k odd, we use (ii) to reduce to v1 ∧ v2 ∧···∧vn + v2 ∧ v1 ∧···∧vn = 0. The last assertion follows by using these equalities to reduce every quasi-zero until reaching 0. Definition 2.10. The standard Grassmann semialgebra V with respect to a given generating set {bi : i ∈ I} of V , also denoted G(V ), is T (V )/Φ, where Φ is the congruence generated by (v ∧ v, 0), ∀v ∈ V. Accordingly G(V )k is T (V )k/Φ and G(V )≥2 = T (V )≥2/Φ. ≥ T − T The standard Grassmann triple is (G(V ) 2, G(V )≥2 , ( )), where G(V )≥2 is theproductofelementsofV of length ≥ 2, and (−) is as in Theorem 2.4. 2.1. Symmetrization and the twist action. There is a general way to provide a negation map for arbitrary Grassmann semialgebras. Although T -modules ini- tially may lack negation, one can obtain negation maps for them through the next main idea, the symmetrization process, which although a special case of super- semialgebras and their modules, provides a crucial method of creating a triple. Tropical symmetrization dates back to [7], and we recall the treatment for systems from [16]. GRASSMANN SEMIALGEBRAS AND THE CAYLEY-HAMILTON THEOREM 191 Definition 2.11. Given any T -monoid module M, define its Z2-graded sym- metrization M = M×M, with componentwise addition. Also define T =(T×{0}) ∪ ({0}×T)withthetwist action of T on M given by the super-action, namely (2.3) (a0,a1) ·tw (c0,c1)=(a0c0 + a1c1,a0c1 + a1c0),ai ∈T,ci ∈M. Definition 2.12. The switch map (−)sw on the symmetrized module M is given by (−)sw(c0,c1)=(c1,c0). If A is a semiring containing T , then we define the twist action as in (2.3), but this time with ai,ci ∈A. Theorem 2.13 ([15, Theorems 2.41, 2.43]). For any T -module A, we can embed A into A via b → (b, 0), thereby obtaining a faithful functor from the category of semirings into the cate- gory of semirings with a negation map (and preserving additive idempotence). This also yields a faithful functor from ordered semigroups to signed (−)sw-bipotent sys- tems. Any A-module M yields a A-module M = M⊕M, which has a signed decomposition where M+ is the first component. This applies to the Grassmann semialgebra: Theorem 2.14. Define G(V ) to be G(V ) modded out by the congruence generated ∼ by (v ∧ v , 0) = (0,v ∧ v) for all v, v ∈ V ,andT to be the corresponding image of T . There is a triple (G(V ) , T , (−)sw), together with an embedding of triples ≥ T − → T − (G(V ) 2, G(V )≥2 , ( )) (G(V ) , , ( )sw) given by c → (c, 0). Proof. Take A = G(V ) in Theorem 2.13. (−)(v ∧ v )=v ∧ v → (v ∧ v, 0) = (−)sw(0,v ∧ v)=(−)sw(v ∧ v , 0). Thus (−) matches by (2.13), so the Grassmann relations match. 2.2. The partially reduced Grassmann system (when V is free). The main results of this paper involve the free module V with base B = {b0,...,bn−1}, in the sense that any element of V can be written uniquely as an A-linear combination (n) of the bi.LetVn := A be the free module over the semiring A with basis b := {b0,...,bn−1} of n elements. When V = Vn, this includes the definition in [8, Definition 3.1.2], in which (−) is the identity map. In this work we have two candidates for the Grassmann algebra, given respectively in Theorems 2.14 and 2.22. Remark 2.15. ∧···∧ − π ∧···∧ (i) By Theorem 2.9, bπ(i1) bπ(it) =( ) bi1 bit for any permutation π. Also, every simple tensor in which some bi repeats is 0. n (ii) Gk is free with a base of 2 k elements. For instance G2(V3)hasbase b1 ∧ b2, b1 ∧ b3, b2 ∧ b3 and their “negations” b2 ∧ b1, b3 ∧ b1, b3 ∧ b2.This phenomenon gives rise to the “eigenvalue pair” of §3.2 below. 192 LETTERIO GATTO AND LOUIS ROWEN Lemma 2.16. For the free Grassmann semialgebra, G = Geven ⊕ Godd is a super- ≥2 ≥2 ⊕ ≥2 semialgebra, and its ideal G = Geven Godd has the negation map from Theo- rem 2.4. Proof. By linearity, we need only check products of the bi. Lemma 2.17. (−) is well-defined, and ¯ ∧···∧¯ − π¯ ∧···∧¯ ∀ ≥ bπ(i1) bπ(it) =( ) bi1 bit , t 2. Proof. (−) is well-defined by Theorem 2.4. The formula follows from writing a permutation as the product of transpositions, noting that the sign of a permutation is well-defined, and counting the number of times (−) occurs. Lemma 2.18. Suppose α b ⊗···⊗b + d α b ⊗···⊗b + d , where i i i1 ik i i i1 ik ◦ i < ··· Proof. Match components, eliminating those components in which some bi repeats or some of the bi descend. Definition 2.19. T (V )doub is the ideal of T (V ) generated by all elements bi ⊗ bi for all i. ± ···⊗ Lemma 2.20. Any nonzero element of T (V ) is a sum of terms ( )αbi1 bik +d, where i1 < ··· Proof. We rearrange the bi appearing in the summands, noting that any time a bi repeats, the product is in T (V )doub. We use Lemma 2.18 to avoid T (V )doub in our computations. We will need the following nondegeneracy result. (n) Proposition 2.21. Suppose V = A and u, u ∈ G(V )k for 2 ≤ k Theorem 2.22. More in line with [8], define the partially reduced Grassmann ♦ ∧ algebra G(V )≥2 to be T (V )≥2 modded out by the congruence generated by (bi bi, 0) ∼ for all i, G(V ) to be T (V ) modded out by the congruence generated by (bi ∧bj, 0) = (0,bj ∧ bi) for all i, j,andT to be the corresponding image of T . There is a triple (G(V ) , T , (−)sw), together with an embedding of triples (G(V )♦ , T , (−)) → (G(V ), T , (−) ) ≥2 G(V )≥2♦ sw given by c → (c, 0). GRASSMANN SEMIALGEBRAS AND THE CAYLEY-HAMILTON THEOREM 193 ♦ Note 2.23. G(V )≥2 is clearly a homomorphic image of G(V )≥2, since it has more ⊗ ⊗ ♦ relations. (All v v are sent to 0,notjustthebi bi.) Although G(V )≥2 and G(V )≥2 both coincide with the Grassmann algebra in the classical setting where V is a vector space over a field, they differ in the semiring setting, since bi ⊗bj +bj ⊗bi is not 0 in G(V )≥2. 2.3. Digression: Related notions. For the remainder of this section we examine algebraic notions related to this paper, even though one can bypass them for the proofs of Theorems 3.14 and 3.15. 2.3.1. ThecasewhenV already has a negation map. We have seen that V itself need not have a negation map, for us “almost” to define a negation map on T (V ). In case V does have a negation map (−),2 we need a slight modification. We define a negation map on the tensor product V ⊗ W by (−)(v ⊗ w)=((−)v) ⊗ w. When W also has a negation map (−) we define a negated tensor product V ⊗(−) W by imposing the extra axiom ((−)v) ⊗(−) w = v ⊗(−) ((−)w),v∈ V,w ∈ W. (One mods out the tensor product by the congruence generated by all elements ((−)v ⊗ w, v ⊗ (−)w).) Remark 2.24. The appropriate triple is (G, TG, (−)), where TG = {v1 ∧···∧vt : vi ∈ V, t ∈ N}, the submonoid generated by T ,with(−)(v1 ∧ ··· ∧ vt)= ((−)v1) ∧···∧vt. 2.3.2. Comparison with [8]. The arguments of [8], whose objective is to obtain a Grassmann algebra point of view for Pl¨ucker relations, can be adapted to this situation. We use the systemic version which enables us to replace the “bend relation” f ∼ g of [8] for f,g ∈ Hom(V,A)byf + g beingaquasi-zerointhe ◦ sense that (f + g)(v) ∈A for every v ∈ V. Then the Pl¨ucker relations in [8, Proposition 4.1.2] become the conditions that i∈A\B vA−{i}vB+{i} is a quasi-zero. Note 2.25. The flavor of the Grassmann algebra might be better preserved by taking the negation map (−) not to be the identity map, but rather as defined here, which also could be obtained using symmetrization. Note also that G is commutative in [8, Definition 3.1.2]. So why does [8, Definition 3.1.2] work? The answer is that G is largely a book-keeping devise to keep track of sets of vectors without repetition, and application of its theory to matroids does not require much of multiplication other than ei ∧ ei = 0. [8, Proposition 3.1.4] is formal. One needs a cancelation result parallel to [8, Lemma 3.2.2], and [8, Proposition 4.2.1] requires the ability to switch vectors ei and ej . Remark 2.26. Suppose that A is “zero sum free” in the sense that a1+a2 = 0 implies a1 = a2 = 0. Then the base B of a free module V is unique up to multiplication of invertible elements of A. (Otherwise some bi does not appear in the new base, and we cannot recover bi since we cannot zero out extraneous coefficients.) 2 For example V could be the free A-module with a negation map, with base {bi, (−)bi : i ∈ I}. 194 LETTERIO GATTO AND LOUIS ROWEN 2.3.3. Digression: The Grassmann envelope. Remark 2.27. Just as with classical algebra, one can use G to study a super- semialgebra A = A0 ⊕A1 using its Grassmann envelope A0 ⊗ G0 + A1 ⊗ G1 ⊂A⊗G. Following Zelmanov, we say that a super-semialgebra A is super-P if its Grassmann envelope is P. For example, A is super-commutative if its Grassmann envelope is commutative. In particular, G itself is super-commutative. Then one can study linear algebra over super-commutative super-semialgebras, super-anticommutative super-semialgebras, and so forth, as indicated in [21, §8.2.2]. 3. Hasse-Schmidt derivations on Grassmann semi-algebras Having set out the general framework, let us turn to the situation at hand. We review our set-up, in the special case of power series over endomorphisms of the (n) Grassmann algebra. As before, Vn := A is the free module with basis b := {b0,...,bn−1}. (We start our subscripts with 0 in consonance with the notation for projective space.) Let T0(Vn)=A,andTk(Vn) := Vn ⊗Vn ⊗···⊗Vn be its k tensor power. Define a negation (−):T2(Vn) → T2(Vn) by mapping u ⊗ v to v ⊗ u.In particular (−)(u ⊗ u)=u ⊗ u.Weextendthisto(−):Tk(Vn) → Tk(Vn)bymeans of Theorem 2.4 and Lemma 2.17. Let T≥2(Vn)={0}∪ Tk(Vn), k≥2 a semiring†† with multiplication given by tensoring. We consider two variants of the Grassmann algebra: I (1) G(V )≥2, modding out by the congruence of T≥2(Vn) generated by all {(bi ⊗ bi, 0) : 0 ≤ i r Vn = Vn, where r≥0 0 1 r T (V ) V = A, V = V , and V := r n for r ≥ 2. n n n n I∩T (V ) r n Thus u ∧ v denotes the image of u ⊗ v through the natural map T (Vn) → Vn. Here is ◦ . r Remark 3.1. By Theorem 2.4, each submodule Vn, r ≥ 2, inherits a negation map by putting (−)(u1 ∧ u2 ···∧ur)=u2 ∧ u1 ∧···∧ur. Remark 3.2. ≥ r ∧ ∧···∧ (i) For each r 2, Vn is spanned by words bi0 bi1 bir−1 of length r A r r.Inparticular, Vn is a free module spanned by [b]λ,where ≥···≥ r ∧ ∧···∧ λ :=(λ1 λr), [b]λ := bλr b1+λr−1 br−1+λ1 . GRASSMANN SEMIALGEBRAS AND THE CAYLEY-HAMILTON THEOREM 195 r We are interested in the N-graded power series semiring ( Vn)[[z]]:=⊕r≥0 Vnz of Definition 1.14 (and later its super-version), and its endomorphisms. Since the congruences are homogeneous, we define (3.1) r r r ≥1 ≥2 =1 Vn := Vn, Vn := Vn, and Vn := Vn r≥1 r≥2 r=1 { } i ∈ Definition 3.3. Let D z := i≥0 Diz End( Vn)[[z]] be homogeneous of r r degree 0 (i.e. Di( Vn) ⊆ Vn and in particular Di(Vn) ⊆ Vn)).If (3.2) D{z}(u ∧ v)=D{z}u ∧ D{z}v wesaythatitisaHasse-Schmidt (HS) derivation on Vn. To simplify notation let us simply denote the identity map on Vn as “1V ,” also identified with D{z} where D0 = 1 and all other Di = 0. Equation (3.2) is equivalent to: (3.3) Dk(u ∧ v)= Diu ∧ Dj v, ∀k ≥ 0, ∀u, v ∈ Vn. i+j=k r For r ≥ 2, any element of Vn is a linear combination of monomials v1 ∧···∧vr of length r. The definition shows that D{z} is uniquely determined by the values it takes on elements of V . In the following we shall restrict to a special class of HS derivations, useful for the applications. ∈ Proposition 3.4. For any f EndA(Vn), there exists a unique HS-derivation f f i i D {z} on V such that D {z}| = f z . n Vn i≥0 A f { } Proof. For the chosen -basis of the module V we necessarily have D z (bj )= i i i i f { } i≥0 f (bj )z .Writef(z)for i≥0 f z . One defines D z on V by setting for each degree: f { } ∧···∧ ∧···∧ ≤ ≤ (3.4) D z (bi1 bij )=f(z)bi1 f(z)bij , 1 j r. If D were another derivation satisfying the same initial condition, it would coincide on all the basis elements of Vn, which generate all elements of Vn. f ∧ Example 3.5. LetuscomputeD2 (b1 b2)wheref(bi)=bi+1.Then f ∧ f ∧ f ∧ f ∧ f D2 (b1 b2)=D2 (b1) b2 + D1 b1 D1 b2 + b1 D2 b2 2 2 = f (b1) ∧ b2 + f(b1) ∧ f(b2)+b1 ∧ f (b2) = b3 ∧ b2 + b2 ∧ b3 + b1 ∧ b4 b1 ∧ b4, since b3 ∧ b2 + b2 ∧ b3 is a quasi-zero. From now on we shall fix the endomorphism f once and for all, and writeD{z} := f i D (z)andD := D1|V := f.AlsowewriteDiv for Di(v)andD{z}v for Div · z . ∈ i i In particular, for each v Vn the equality Div = D1v = f (v)holds. Lemma 3.6. For u, v ∈ Vn, (i) D{z}v = v + D{z}(D1v)z. (ii) D{z}(u ∧ v)=u ∧ D{z}v + zD{z}(D1u ∧ v). 196 LETTERIO GATTO AND LOUIS ROWEN Proof. { } i i−1 { } (i) D z v = v + i≥1 Divz = v + i≥1(Di−1D1vz )z = v + D z (D1v)z. (ii) D{z}(u ∧ v)=D{z}u ∧ D{z}v =(u + zD{z}D1u) ∧ D{z}v) = u ∧ D{z}v + zD{z}D1u ∧ D{z}v = u ∧ D{z}v + zD{z}(D1u ∧ v). 3.1. The canonical quasi-inverse of D{z}. { } i ∈ =1 Definition 3.7. D z := i≥0 Diz End( Vn)[[z]] is a (left) quasi-inverse of D{z} if =1 (3.5) D{z}D{z}u u, ∀u ∈ Vn. Our next task consists in constructing a quasi-inverse D{z} of the HS derivation D{z}, that we will achieve through a number of steps necessary to cope with the difficulty of not having a natural negation map on Vn. This can be done in two ways: First do it in the classical case, and then apply the “transfer principle” of Remark 1.7. However, one gets more precise information by taking the direct analog. Construction. Towards this purpose we first consider the map: D : Vn −→ EndA( Vn) (3.6) u −→ Du ≥2 such that Du(v) := v ∧ D1u for all v ∈ Vn.Ifv ∈ Vn, we may assume it is of the form v = v1 ∧ v2 with v1 ∈ Vn. In this case we define (3.7) Du(v)=Du(v1 ∧ v2)=Du(v1) ∧ v2 = v1 ∧ D1u ∧ v2. It is easily seen that (3.7) suffices to define Du on all the Grassmann semi-algebras and also that the definition does not depend on the representation of the same element v ∈ Vn. In fact any one such is a finite linear combination of tensors of the form v := v1 ∧···∧vk, and the definition of Du(v) does not change if we replace the given expression of v with an equivalent one after an even permutation of the factors. It will be useful to identify Vn as a subset of EndA( Vn), by viewing each of its elements as a (wedge) multiplication operator, under the map v → v∧ , i.e. v(w)=v ∧ w for all w ∈ Vn. { } 2 ··· n → Definition 3.8. Let D z :=1+D1z + D2z + + Dnz : Vn EndA( Vn)[z] k defined as follows. If u1 ∧···∧uk ∈ Vn,then (3.8) D{z}(u1 ∧···∧uk)=D{z}(u1) ◦···◦D{z}(uk) 1 where ◦ is the composition in EndA( Vn) and where for all u ∈ Vn = Vn we set (3.9) (D{z}u)(v)=u ∧ v + zDu(v) ∈ EndA( Vn)[z] ∈ { } ∧ { } acting on v Vn as D z u(v)=u v +(Du)v.NowweextendD z to a map Vn → EndA( Vn) defining it on monomials of degree k ≥ 2via k (1+D1z+···+Dkz )(u1∧···∧uk)=D{z}(u1∧···∧uk)=D{z}(u1)◦···◦D{z}(uk) i Remark 3.9. By definition it follows that if u ∈ Vn,thenDj u =0forallj>i. GRASSMANN SEMIALGEBRAS AND THE CAYLEY-HAMILTON THEOREM 197 Example 3.10. In this example we compute D1(u1 ∧ u2), D2(u1 ∧ u2)and D2(u1 ∧ u2 ∧ u3) to illustrate how the definition works. By definition D1(u1 ∧ u2) ∧ 2 ◦ and D2(u1 u2) are the coefficients of z and z in the expansion (Du1) (D1u2) that we apply to a test element w ∈ Vn. By definition one may assume w ∈ Vn. One has: 2 (u1 ∧ u2 + z(Du1 ◦ u2 + u1 ◦ Du2)+z Du1 ◦ Du2)w 2 = u1 ∧ u2 ∧ w + zDu1(u2 ∧ w)+u1 ∧ Du2(w)+z Du1(Du2(w)) 2 = u1 ∧ u2 ∧ w + z(u2 ∧ D1u1 ∧ w + u1 ∧ (w ∧ D1u2)) + z Du1(w ∧ D1u2) 2 = u1 ∧ u2 ∧ w + z(u2 ∧ D1u1 + D1u2 ∧ u1) ∧ w + z (w ∧ D1u1 ∧ D1u2) 2 =[u1 ∧ u2 +(D1u2 ∧ u1 + u2 ∧ D1u1)z +(D1u1 ∧ D1u2)z ] ∧ w. We have obtained: D1(u1 ∧ u2)=(D1u2 ∧ u1 + u2 ∧ D1u1)∧ : Vn → Vn, D2(u1 ∧ u2)=(D1u1 ∧ D1u2)∧ : Vn → Vn. Similarly we can find that D2(u1 ∧ u2 ∧ u3)=D2(u1 ∧ u2) ◦ u3 + D1(u1 ∧ u2) ◦ D1u3, i.e., more explicitly D2(u1 ∧ u2 ∧ u3) ∧ w = D1u1 ∧ D1u2 ∧ u3 ∧ w +(D1u2 ∧ u1 + u2 ∧ D1u1) ∧ w ∧ D1u3 from which D2(u1 ∧ u2 ∧ u3) ∧ w =(D1u1 ∧ D1u2 ∧ u3 + D1u2 ∧ D1u3 ∧ u1 + u2 ∧ D1u3 ∧ D1u1) ∧ w =(D1u1 ∧ D1u2 ∧ u3 + u1 ∧ D1u2 ∧ D1u3 + D1u1 ∧ u2 ∧ D1u3) ∧ w. ∧ ∧ ∧ ∧ We could say that the operators D1(u1 u2), D2(u1 u2), and D2(u1 u2 u3) are “represented” respectively by the following elements of Vn: D1u2 ∧ u1 + u2 ∧ D1u1 = D1(u2 ∧ u1),D1u1 ∧ D1u2, and D1u1 ∧ D1u2 ∧ u3 + u1 ∧ D1u2 ∧ D1u3 + D1u1 ∧ u2 ∧ D1u3. ∧ Remark 3.11. LetuscheckthatD1u v +(D1u)(v) 0, which is the sense we want to give to the expression D1 + D1 0. For all u, v ∈ Vn.Ifu ∈ Vn and v = v1 ∧ v2 with v1 ∈ Vn: D1u ∧ v +(D1u)(v)=(D1u ∧ v1) ∧ v2 +(Du)(v1) ∧ v2 = D1u ∧ v1 ∧ v2 + v1 ∧ D1u ∧ v2 =(D1u ∧ v1 + v1 ∧ D1u) ∧ v2 0. More in general we have the following crucial: Theorem 3.12. The polynomial D{z} is a quasi-inverse of D{z},inthesense that for all u ∈ Vn (3.10) D{z}D{z}u u. 198 LETTERIO GATTO AND LOUIS ROWEN Proof. We first check that the property holds for all u ∈ Vn. Then, for all w ∈ Vn (D{z}D{z}u)(w)=(D{z}u + zDD{z}u)(w) = D{z}u ∧ w + zw ∧ D1D{z}u { } ∧ · ∧ { } =(u + zD1D z u) w + z w D1D z u = u ∧ w + z D1D{z}u ∧ w + w ∧ D1D{z}u u ∧ w ∈ { } { } ∧ ∧ for all w Vn.Thuswehaveprovedthat D z D z u u ,andthe ∈ 1 property is checked for all u Vn. Now, we argue by induction, by supposing ≤k−1 k the property holds true for all u ∈ Vn. Let us prove it for all u ∈ Vn.In this case we can assume u of the form u1 ∧ v, with u1 ∈ Vn.Then D{z}(D{z}(u1 ∧ v)) = D{z}(D{z}u1) ◦ D{z}(D{z}v) (u1 ∧ v) ∧ having used induction and the first step. Counterexample 3.13. Quite surprisingly, while D{z} is a quasi-inverse of D{z}, the reverse is not true. For instance, for all u, v ∈ Vn one has: D{z}(D{z}u)(v) ∧ { } { } ∈ { } u D z v = u + D z D1u. Let us check it, recalling that if u Vn then D z u = i · i u + i≥0 D1u z . D{z}(D{z}u)(v) = D{z}(u ∧ v + z · v ∧ D1u) (definition (3.7) of(D{z}u)(v)) = D{z}u ∧ D{z}v + zD{z}v ∧ D{z}D1u (D{z}is a HS derivation) = u ∧ D{z}v + zD{z}D1u ∧ D{z}v + zD{z}v ∧ D{z}D1u (using D{z}u = u + D{z}D1u) u ∧ D{z}v. As an additional check, notice that if D{z} were a quasi-inverse of D{z},then 2 2 (1 + D1z + D2z + ·)(1 + D1 + D2z + ···) 0 In particular, considering the coefficient of z2 in both sides, the following surpassing relation should hold: (3.11) D2 + D1D1 + D2 0. 1 But (3.11) already fails for u ∈ Vn = Vn. Indeed, for all v ∈ Vn, and noting that D2u = 0, by Remark 3.9: (D2u + D1D1u + D2u)(v) = D2u ∧ v + D1(v ∧ D1u) (applying to a test vector v) = D2u ∧ v + D1v ∧ D1u + v ∧ D2u (Leibniz rule enjoyed by D1) = D1v ∧ D1u + D2u ∧ v + v ∧ D2u D1v ∧ D1u Theorem 3.14. D{z}(D{z}u ∧ v) u ∧ D{z}v, ∀u, v ∈ Vn. k Proof. Suppose u, v are homogeneous, say u ∈ Vn and v ∈ Vn. Then, by Definition 3.8 of D{z}, D{z}(D{z}u ∧ v)=D{z}(D{z}u) ◦ D{z}v GRASSMANN SEMIALGEBRAS AND THE CAYLEY-HAMILTON THEOREM 199 By Theorem 3.12, D{z}(D{z}u∧v)(w) u◦(D{z}v)(w)=u∧(D{z}v)(w), because u acts as an endomorphism on vectors of Vn as u ∧ , for all w ∈ Vn. 3.2. The Cayley-Hamilton formulas for semialgebras. Formally define ζ = b0 ∧ b1 ∧···∧bn−1 and ζ = b1 ∧ b0 ∧···∧bn−1. Thus ζ =(−)ζ, and ∈A (3.12) Diζ = eiζ + eiζ ,ei,ei . In other words, (ei,ei) could be called the eigenvalue pair of Di restricted to n Vn (where in some sense ei is the negated part). Let En(z) be the eigenvalue polynomial of D{z}, i.e. { } { } ··· n ··· n En(z)ζ := D z ζ + D z ζ =(1+e1z + + enz )ζ +(1+e1z + + enz )ζ . { } { } In particular if one sets Diζ = hiζ + hiζ , the relations D z D z ζ ζ and D{z}D{z}ζ ζ yield the relation ··· ··· (3.13) (hn + e1hn−1 + + en)+(hn + e1hn−1 + + en) 0. Theorem 3.15. The Cayley-Hamilton formulas (3.14) ··· ∧ − ··· ∧ ((Dnu + e1Dn−1u + + enu) v)( )((Dnu + e1Dn−1u + + enu) v) 0 >0 hold for all u ∈ Vn, i.e., the left side is a quasi-zero. n−i Proof. If u = ζ the theorem is true, due to (3.13). Then assume that u ∈ Vn, for some 1 ≤ i ≤ n − 1. This follows from the transfer principle of Remark 1.7, since the assertion was proved (with equality) for classical algebras in [6], and all the extra quasi-zeros appear in the right. But we also would like to give a direct i proof. For all v ∈ Vn we have the surpassing relation (3.14). Matching degrees yields the surpassing relation between the n-th degree coefficient of the left side and the n-th degree coefficient of the right side of (3.14) which is: Dnu ∧ v + D1(Dn−1u ∧ v)+···+ Dn(u ∧ v) u ∧ Dnv. Since D{z}v is a polynomial of degree at most i (3.15) ··· ∧ − ··· ∧ ((Dnu + e1Dn−1u + + enu) v)( )((Dnu + e1Dn−1u + + enu) v)=0. n − n−1 ··· − ∈ >0 Corollary 3.17. (D1 +(e1( )e1)D1 + +(en( )en))u 0 for all u Vn, − n−i n−i − n−i where we interpret (ei( )ei)Di (u) to be eiDi u ( ) eiDi u. Proof. By Theorem 3.15, ((Dnu + e1Dn−1u + ···+ enu) ∧ v)(−)((Dnu + e Dn−1u + ···+ e u) ∧ v) 0. 1 n But D{z} is by hypothesis the unique HS-derivation on Vn associated to the i endomorphism D1 (see Proposition 3.4). In particular Diu = D1u. Note 3.18. When working with G(V )≥2, we obtain equality in Theorem 3.15 and Corollary 3.17 since the only quasi-zeros are 0, by Theorem 2.9. 200 LETTERIO GATTO AND LOUIS ROWEN References [1] Marianne Akian, St´ephane Gaubert, and Alexander Guterman, Tropical Cramer determi- nants revisited, Tropical and idempotent mathematics and applications, Contemp. Math., vol. 616, Amer. Math. Soc., Providence, RI, 2014, pp. 1–45, DOI 10.1090/conm/616/12324. MR3221324 [2] George M. Bergman, The diamond lemma for ring theory, Adv. in Math. 29 (1978), no. 2, 178–218, DOI 10.1016/0001-8708(78)90010-5. MR506890 [3] Letterio Gatto, Schubert calculus via Hasse-Schmidt derivations,AsianJ.Math.9 (2005), no. 3, 315–321, DOI 10.4310/AJM.2005.v9.n3.a2. MR2214955 [4] Letterio Gatto and Parham Salehyan, Hasse-Schmidt derivations on Grassmann algebras: With applications to vertex operators, IMPA Monographs, vol. 4, Springer, [Cham], 2016. MR3524604 [5]L.GattoandI.Scherbak,On generalized Wronskians, Contributions in Algebraic Geome- try, Impanga Lecture Notes (P. Pragacz Ed.), EMS Congress Series Report, 257–296, 2012, available at http://xxx.lanl.gov/pdf/1310.4683.pdf. [6] Letterio Gatto and Inna Scherbak, Hasse-Schmidt derivations and Cayley-Hamilton theorem for exterior algebras, Functional analysis and geometry: Selim Grigorievich Krein centen- nial, Contemp. Math., vol. 733, Amer. Math. Soc., Providence, RI, 2019, pp. 149–165, DOI 10.1090/conm/733/14739. MR3985273 [7] S. Gaubert, Th´eorie des syst`emes lin´eaires dans les diodes,Th`ese, Ecole´ des Mines de Paris, 1992. [8] Jeffrey Giansiracusa and Noah Giansiracusa, A Grassmann algebra for matroids,Manuscripta Math. 156 (2018), no. 1-2, 187–213, DOI 10.1007/s00229-017-0958-z. MR3783573 [9] Jonathan S. Golan, The theory of semirings with applications in mathematics and theoret- ical computer science, Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 54, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1992. MR1163371 [10] F. K. Schmidt and H. Hasse, Noch eine Begr¨undung der Theorie der h¨oheren Differen- tialquotienten in einem algebraischen Funktionenk¨orper einer Unbestimmten. (Nach einer brieflichen Mitteilung von F.K. Schmidt in Jena) (German), J. Reine Angew. Math. 177 (1937), 215–237, DOI 10.1515/crll.1937.177.215. MR1581557 [11] Nickolas Heerema, Derivations and embeddings of a field in its power series ring, Proc. Amer. Math. Soc. 11 (1960), 188–194, DOI 10.2307/2032953. MR123568 [12] Nickolas Heerema, Derivations and embeddings of a field in its power series ring. II, Michigan Math. J. 8 (1961), 129–134. MR136601 [13] Nickolas Heerema, Higher derivations and automorphisms of complete local rings, Bull. Amer. Math. Soc. 76 (1970), 1212–1225, DOI 10.1090/S0002-9904-1970-12609-X. MR266916 [14] Zur Izhakian, Manfred Knebusch, and Louis Rowen, Supertropical quadratic forms I,J.Pure Appl. Algebra 220 (2016), no. 1, 61–93, DOI 10.1016/j.jpaa.2015.05.043. MR3393451 [15] J. Jun, K. Mincheva, and L. Rowen, Homology of module systems, arXiv:1809.01996,J.Pure Appl. Algebra, to appear (2020). [16] Jaiung Jun and Louis Rowen, Categories with negation, Categorical, homological and com- binatorial methods in algebra, Contemp. Math., vol. 751, Amer. Math. Soc., Providence, RI, [2020] c 2020, pp. 221–270, DOI 10.1090/conm/751/15120. MR4132092 [17] E. B. Kacov, The tensor product of functors (Russian), Sibirsk. Mat. Zh. 19 (1978), no. 2, 318–327, 478. MR489991 [18] Yefim Katsov, Tensor products and injective envelopes of semimodules over additively regular semirings, Algebra Colloq. 4 (1997), no. 2, 121–131. MR1682765 [19] Oliver Lorscheid, The geometry of blueprints: Part I: Algebraic background and scheme the- ory, Adv. Math. 229 (2012), no. 3, 1804–1846, DOI 10.1016/j.aim.2011.12.018. MR2871157 [20] M. H. A. Newman, On theories with a combinatorial definition of “equivalence.”, Ann. of Math. (2) 43 (1942), 223–243, DOI 10.2307/1968867. MR7372 [21] L. H. Rowen, Algebras with a negation map, 75 pages, arXiv:1602.00353 [math.RA]. [22] Louis Rowen, An informal overview of triples and systems,Rings,modulesandcodes, Contemp. Math., vol. 727, Amer. Math. Soc., Providence, RI, 2019, pp. 317–335, DOI 10.1090/conm/727/14644. MR3938159 GRASSMANN SEMIALGEBRAS AND THE CAYLEY-HAMILTON THEOREM 201 [23] Michihiro Takahashi, On the bordism categories. I, Math. Sem. Notes Kobe Univ. 8 (1980), no. 3, 527–546. MR615872 [24] Wikipedia, Exponential of nilpotent derivation with divided Leibniz condition powers is en- domorphism. Dipartimento di Scienze Matematiche, Politecnico di Torino, C. so Duca degli Abruzzi 24, 10129 Torino - Italia Email address: [email protected] Department of Mathematics, Bar-Ilan University, Ramat-Gan 52900, Israel Email address: [email protected]